A new theorem on exponential stability of periodic evolution families on Banach spaces
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Date
2003-02-11
Authors
Buse, Constantin
Jitianu, Oprea
Journal Title
Journal ISSN
Volume Title
Publisher
Southwest Texas State University, Department of Mathematics
Abstract
We consider a mild solution v<sub>f</sub> (·, 0) of a well-posed inhomogeneous Cauchy problem v̇(t) = A(t)v(t) + ƒ(t), v(0) = 0 on a complex Banach space X, where A(·) is a 1-periodic operator-valued function. We prove that if vƒ (·, 0) belongs to AP0 (ℝ₊, X) for each ƒ ∈ AP0(ℝ₊, X) then for each x ∈ X the solution of the well-posed Cauchy problem u̇(t) = A(t)v(t), u(0) = x is uniformly exponentially stable. The converse statement is also true. Details about the space AP0(ℝ₊, X) are given in the section 1, below. Our approach is based on the spectral theory of evolution semigroups.
Description
Keywords
Almost periodic functions, Exponential stability, Periodic evolution families of operators, Integral inequality, Differential inequality on Banach spaces
Citation
Buse, C., & Jitianu, O. (2003). A new theorem on exponential stability of periodic evolution families on Banach spaces. <i>Electronic Journal of Differential Equations, 2003</i>(14), pp. 1-10.
Rights
Attribution 4.0 International